Percolation theory cluster structure

It is clear that the exponent value is not related to the fractal dimension and to percolation theory. The structure and the connections of a gel are the result of a sequence of different processes gelation, aging and shrinkage. The a value should describe the way the clusters are connected between them and not the structure inside the clusters. 

Objective criteria which clearly distinguish cluster structures from cellular and mesh structures have been developed by S. F. Shandarin (1983) [65] on the basis of percolation theory, whose usefulness for differentiating structures was conjectured by Ya.B. The percolation of cellular- or mesh-structure objects on a union of r-neighborhoods begins to occur at smaller r than in the case of a system of independent, randomly located points or clusters with the same (average) density. 

The percolation probability has different values based on the classical theory site or bond percolation for different structures, as shown in Table 12.2. This critical percolation volume fraction, <, is calculated from the percolation threshold and the space filling factor. The volume fraction for site percolation for various structures is essentially the same as follows. In three dimensions, the site percolation threshold occurs at —16% volume. Near the percolation threshold the average cluster size diverges as does the spanning length of clusters. 

Figure 2 also shows that when 62—63% of H bonds are broken, the piece of ice is disintegrated into small separate clusters, and the network of H bonds is completely broken down. This result is slightly different from thaU (60—61%) obtained by assuming equal probability of rupture of all H bonds. Let us note 0 3i 60—61% is also the threshold provided by the percolation theory for the tetrahedral structure. 

A model for ionic clustering in "Nafion" (registered trademark of E. I. du Pont de Nemours and Co.) perfluorinated membranes is proposed. This "cluster-network" model suggests that the solvent and ion exchange sites phase separate from the fluorocarbon matrix into inverted micellar structures which are connected by short narrow channels. This model is used to describe ion transport and hydroxyl rejection in "Nafion" membrane products. We also demonstrate that transport processes occurring in "Nafion" are well described by percolation theory. 

Crete surface to the bulk of the concrete. Permeability is high (Figure 1.6) and transport processes like, e. g., capillary suction of (chloride-containing) water can take place rapidly. With decreasing porosity the capillary pore system loses its connectivity, thus transport processes are controlled by the small gel pores. As a result, water and chlorides will penetrate only a short distance into concrete. This influence of structure (geometry) on transport properties can be described with the percolation theory [8] below a critical porosity, p, the percolation threshold, the capillary pore system is not interconnected (only finite clusters are present) above p the capillary pore system is continuous (infinite clusters). The percolation theory has been used to design numerical experiments and apphed to transport processes in cement paste and mortars [9]. 

The critical indices estimated from these relations fall into the admissible ranges of variation P = 0.39-0.40, V = 0.8-0.9, and t = 1.6-1.8, determined in terms of the percolation model for three-dimensional systems. The researchers [7] noted that not only numerical values but also the meanings of these values coincide. Thus the index P characterises the chain structure of a percolation cluster. The 1/p value, which serves as the index of the first subset of the fractal percolation cluster in the model considered [7], also determines the chain structure of the cluster. The index v is related to the cellular texture of the percolation cluster. The 2/df index of the second subset of the fractal percolation cluster is also associated with the cellular structure. By analogy, the index t defines the large-cellular skeleton of the fractal percolation cluster. The relationship between the critical percolation indices and the fractal dimension of the percolation cluster for three-dimensional systems and examples of determination of these values for filled polymers are considered in more detail in the book cited [7]. Thus, these critical indices are universal and significant for analysis of complex systems, the behaviour of which can be interpreted in terms of the percolation theory. 

Catalyst deactivation (e.g. deposition of coke as a function of time) changes the porous structure with time. At a certain time the pore network cannot be passed by the reactant gas. This is the so-called percolation threshold [52]. Before this percolation threshold appears pore clusters of different size may be found within the network and at least one cluster ranges over the entire pellet. Percolation theory deals with the number and properties of these clusters. 

Percolation phenomena deal with the effect of clustering and coimectivity of microscopic elements in a disordered medium [129], Percolation theory represents a random composite material as a network or lattice structure of two or more distinct types of microscopic elements or phase domains, the so-called percolation sites. These elements represent mutually exclusive physical properties, e.g., electrically conducting vs. isolating phase domains, pore space vs. solid matrix, atoms with spin up vs. spin down states. Here, we will refer to black and white elements for definiteness. The network onto which black and white elements of the composite medium are distributed could be continuous (continuum percolation) or discrete (discrete or lattice percolation) it could be a disordered or regular network. With a probability p a randomly chosen percolation site will be... 

Combination of the macrohomogeneous approach for porous electrodes with a statistical description of effective properties of random composite media rests upon concepts of percolation theory (Broadbent and Hammersley, 1957 Isichenko, 1992 Stauffer and Aharony, 1994). Involving these concepts significantly enhanced capabilities of CL models in view of a systematic optimization of thickness, composition, and porous structure (Eikerling and Komyshev, 1998 Eikerling et al., 2004). The resulting stmcture-based model correlates the performance of the CCL with volumetric amounts of Pt, C, ionomer, and pores. The basis for the percolation approach is that a catalyst particle can take part in reaction only if it is connected simultaneously to percolating clusters of carbon/Pt, electrolyte phase, and pore space. Initially, the electrolyte phase was assumed to consist of ionomer only. However, in order to properly describe local reaction conditions and reaction rate distributions, it is necessary to account for water-filled pores and ionomer-phase domains as media for proton transport. 

At a critical value of the fraction of objects of one type, these objects would form an extended cluster that connects the opposite external faces of the sample. At this so-called percolation threshold, the corresponding physical property represented by the connected objects would start to increase above zero. Thereby percolation theory establishes constitutive relations between composition and structure of heterogeneous media and their physical properties of interest. For porous electrodes or catalyst layers in PEFC, these properties are electrical conductivities of electrons and protons, diffiisivities of gaseous reactants and water vapor, and liquid water permeability. 

Percolation theory rationalizes sizes and distribution of connected black and white domains and the effects of cluster formation on macroscopic properties, for example, electric conductivity of a random composite or diffusion coefficient of a porous rock. A percolation cluster is defined by a set of connected sites of one color (e.g., white ) surrounded by percolation sites of the complementary color (i.e., black ). If p is sufficiently small, the size of any connected cluster is likely to be small compared to the size of the sample. There will be no continuously connected path between the opposite faces of the sample. On the other hand, the network should be entirely connected if p is close to 1. Therefore, at some well-defined intermediate value of p, the percolation threshold, pc, a transition occurs in the topological structure of the percolation network that transforms it from a system of disconnected white clusters to a macroscopically connected system. In an infinite lattice, the site percolation threshold is the smallest occupation probability p of sites, at which an infinite cluster of white sites emerges. 

The first models, describing elastic behavior of fractal structures, were used, as a rule, for simulation within the fimneworks of percolation theory [1-5], Nonhomogeneous statistical mixture of solid and liquid then only displays solid body properties (e g., not equal to zero shear modulus G), when solid component forms percolation cluster, like at gelation in pol5uner solutions. If liquid component there is replaced by vacuum, then bulk modulus. B will also be equal to zero below percolation threshold [1]. Such model gives the following relationship for elastic constants [1,3] ... 

The polymers physical aging represents itself the structure and properties change in time and is the reflection of the indicated materials thermodynamically nonequilibriiun nature [61, 62], As a rule, the physical aging results to polymer materials brittleness enhancement and therefore, the ability of structural characteristics in due course prediction is important for the period of estimation of pol5mier products safe exploitation. For cross-linked polymers the quantitative estimation of structure and properties changes in physical aging process was conducted in Refs. [63, 64] within the frameworks of fracture analysis [65] and cluster model of polymers amorphous state structure [7, 66]. The authors of Ref. [67] use the indicated theoretical models for the description of PC physical aging. Besides, for PC behavior closer definition in the indicated process such theoretical notions were drawn as structure quasiequilibrium state [68] and the thermal cluster model [69], which is one from variants of percolation theory. 

Hence, the condition 3 V 0, which follows from the percolation theory [39] and experimental data for cluster structure of polymers [40, 41], also defines the fractality of the structure of polymers. It is obvious that the indicated condition and, consequently, the fractality of the structure of polymers define the dependence of on the testing temperature, i.e., local order of thermal fluctuation nature. In other words, the structure fractality and local order frozen lower than are interexcluded notions. 

The first models describing the elastic behaviour of fractal structures used, as a rule, simulation within the frameworks of percolation theory [21-25]. Anon-homogeneous statistical mixture of solid and liquid displays solid properties (for instance, shear modulus G not equal to zero) only, when the solid component forms a percolation cluster at gelation in polymer solutions. If the liquid component is replaced by a vacuum then the bulk modulus K. will also be equal to zero below the percolation threshold [21]. This model gives the following relationship for elastic constants [21] ... 

The approaches considered allow modeling of the primary texture of PS and the processes, limited by individual PBUs that mainly correspond to level III and partially to level IV in the hierarchical system of models (see Section 9.6.3). PBUs are identical in regular PSs, and simulation of numerous processes may be reduced to analysis of a process in a single PBU/C or PBU/P. An accurate modeling of the processes in irregular PSs requires the studies of the properties of structure and properties of the ensembles (clusters) of particles and pores (level IV of the system of models) and the lattices of such clusters (levels V to VII of the system of models). Let us consider the composition of clusters on the basis of fractal [127], and the lattices on the basis of percolation [8] theories. 

Proton conductivities of 0.1 S cm at high excess water contents in current PEMs stem from the concerted effect of a high concentration of free protons, high liquid-like proton mobility, and a well-connected cluster network of hydrated pathways. i i i i Correspondingly, the detrimental effects of membrane dehydration are multifold. It triggers morphological transitions that have been studied recently in experiment and theory.2 .i29.i ,i62 water contents below the percolation threshold, the well-hydrated pathways cease to span the complete sample, and poorly hydrated channels control the overall transports ll Moreover, the structure of water and the molecular mechanisms of proton transport change at low water contents. 

There has been no direct verification of the conceptual structure of the theory. That is, a microscopic determination of the cluster distribution function has not been made, and the effects of percolation have not been seen. Assuming that the structure of the glass is well-defined liquidlike clusters in a denser solidlike background, one might expect to be able to see these clusters by either neutron or X-ray scattering. Since v is probably between 100 and 400 A and r c SO at Tscattered wave vectors on the order of 0.1 A could be used. 

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